WASHINGTON — One week after Rob Porter, his staff secretary, resigned amid spousal abuse allegations, President Trump said on Wednesday that he was “totally opposed to domestic violence,” his first condemnation of the alleged conduct behind a scandal that has engulfed the White House.

His statement, which members of both parties had said was long overdue, came as John F. Kelly, Mr. Trump’s chief of staff, faced new questions about his handling of Mr. Porter’s case, including how he could have held a temporary high-level security clearance for more than a year in light of the allegations, and as committees in both the House and the Senate announced they would investigate the circumstances surrounding the granting of Mr. Porter’s clearance.

Mr. Kelly told senior aides last fall to put an immediate end to granting new interim security clearances like the one given to Mr. Porter and directed them to resolve any issues preventing employees who held them at the time from receiving a full clearance, according to two people familiar with the discussion.

At a meeting in the West Wing, Mr. Kelly said he was assigning Kirstjen Nielsen, then his deputy, to enforce the new policy, the people said. But it is not known whether Mr. Kelly, Ms. Nielsen or any other senior officials sought to delve into why Mr. Porter was operating with only an interim clearance.

Mr. Porter resigned last week after allegations that he had abused his two former wives were reported by The Daily Mail. In the following days, Mr. Trump repeatedly declined to comment publicly on the episode — or on the issue of domestic violence — even as he went out of his way to praise Mr. Porter and express sympathy with him, noting that he had denied the accusations.

But during a tax event on Wednesday at the White House, the president finally weighed in, saying: “I’m totally opposed to domestic violence of any kind. Everyone knows that, and it almost wouldn’t even have to be said.”

The F.B.I. learned about the allegations by the two women soon after Mr. Trump was inaugurated last year. But spokesmen and spokeswomen for the White House have insisted that no senior White House officials knew of them until last week. They have said that the career government employees at the White House Personnel Security Office who processed the clearances did not tell them about the allegations uncovered by the F.B.I.

“I am interested in how someone with credible allegations of domestic abuse, plural, can be hired,” said Representative Trey Gowdy, Republican of South Carolina and the chairman of the Oversight and Government Reform Committee, which will look into Mr. Porter’s clearance and the granting of interim clearances generally. He called domestic abuse “a particularly insidious crime” that bears serious consideration in both the hiring and clearance process.

Senator Charles E. Grassley, Republican of Iowa and the chairman of the Judiciary Committee, said that he had instructed his staff to also begin looking into the clearance process and into whether the proper protocols were followed at the White House, while six Democratic senators said that they were concerned about whether there had been “any mishandling of classified information” because of the interim clearances.

The six senators asked Christopher A. Wray, the F.B.I. director, for the names of other employees at the White House who were working “without being able to obtain a permanent security clearance.”

Mr. Porter was allowed to continue serving in his post with an interim clearance while awaiting approval from the Personnel Security Office for a permanent clearance. The little-known office of about a dozen employees, on the fifth floor of the New Executive Office Building down the block from the White House, receives background information from the F.B.I. and determines whether officials should be cleared to have access to sensitive information.

Mr. Wray testified on Tuesday that the F.B.I. had sent the White House a preliminary background report on Mr. Porter in March, a full investigation in July and a more detailed accounting in November. The agency closed its inquiry in January.

But White House officials say the security office, where only one out of three officials who had to sign off on granting Mr. Porter’s clearance had weighed in, never made a final recommendation.

When senior White House aides have been flagged for concern during prior administrations, that information was typically sent to the general counsel at the Office of Administration, which oversees the security office, who would quickly pass the information to the White House counsel or the deputy chief of staff, according to people familiar with the process.

Several former White House officials and people familiar with the security clearance process argue that it is highly unusual and dangerous for people at Mr. Porter’s level to do their jobs indefinitely without the proper clearance.

“The presumption is not that you get clearance; the presumption is actually that you do not get clearance, and you have to prove yourself worthy of having access to vital information,” said Max Stier, the president and chief executive of the Partnership for Public Service, a nonprofit organization that specializes in federal government management issues.

“Interim clearance is simply to allow people to operate for a short time in their jobs while they await a full check, but if you get red flags in that interim clearance process, then it means that you shouldn’t have that interim clearance,” he added.

Even as Mr. Kelly’s past efforts to deal with security clearance issues at the White House were becoming clearer, his shifting public responses to the revelations of Mr. Porter’s past were coming under further scrutiny.

Three people briefed on the situation said that Mr. Kelly learned that the accusations would be published in The Mail last Tuesday, before leaving for a visit to Capitol Hill. In a meeting with a group of aides, including several from the press office, everyone agreed that Mr. Porter would have to resign, the people briefed on the situation said, and a statement from Mr. Kelly was drafted to provide to The Mail.

But Mr. Porter continued to deny the accusations from his former wives. One aide in the discussions pushed back on the belief that Mr. Porter should resign, saying that these were mere allegations, and that if Mr. Porter were forced out over them, other people could be forced from their posts any time an allegation was made. Other aides agreed, and argued for waiting for the story to play out.

At that point, they reached out to Mr. Kelly, who had left for the visit to the Capitol, by phone, the people said, and he said he agreed, telling them to make his statement about Mr. Porter more supportive. Mr. Kelly dictated specific language that he wanted in the statement to Sarah Huckabee Sanders, the White House press secretary.

A short time later, The Mail published Mr. Kelly’s statement calling Mr. Porter “a man of true integrity and honor,” someone with whom he was “proud to serve,” and who had faced “vile” accusations from his former wives.

But soon after the article appeared, Mr. Kelly, who by then had returned to the White House, heard from someone with more detailed knowledge of the allegations against Mr. Porter that more damning information was about to come out, and that the chief of staff should not put himself in the position of being Mr. Porter’s main defender. The people briefed on the discussions would not identify that person.

The conversation prompted Mr. Kelly to go back to Mr. Porter, this time telling him that he “knows what he has to do,” according to those briefed on the discussion.

Mr. Porter agreed to resign, and told his staff that he was stepping down, a White House official said. But the next morning, Wednesday of last week, he told White House aides he wanted to leave on his own terms and help with the transition. Mr. Kelly told him that he had to leave his job, but he agreed to let Mr. Porter attempt a more graceful exit with an unclear departure date, those briefed said.

The scandal has placed Mr. Kelly’s job in jeopardy, leading Mr. Trump to complain privately about him and sound out confidants about potential replacements, including Gary D. Cohn, the director of his National Economic Council, and Representative Kevin McCarthy, Republican of California and the majority leader.

Mr. Trump is said to seem more favorable toward Mr. McCarthy in some of his discussions, seeing him as someone who would be a more willing subordinate than Mr. Cohn might be, according to a person with direct knowledge of the discussions. Yet in other conversations, Mr. Trump has indicated that Mr. Cohn is his pick.

Several of Mr. Trump’s advisers believe the president, who has a long history of quizzing aides about one another behind their backs without taking action, might just be venting. And his interactions with Mr. Kelly have remained mostly positive, according to two West Wing advisers who have witnessed them together.

Still, Mr. Kelly is facing a restive staff that includes some aides who have grown increasingly upset with his handling of the Porter situation, including a constantly shifting set of explanations and timelines that have raised further questions about the White House’s credibility.

As new information about the scandal emerged Wednesday, officials twice postponed, then ultimately canceled, the daily briefing by the press secretary. Press aides told reporters the question-and-answer session had been scrapped in light of a fatal school shooting in South Florida.